The Wisconsin County Coordinate System:

Redefinition of the Foundation

WLIAMarch 2, 2006

Ted Koch – State Cartographer’s OfficeAlan Vonderohe – UW-MadisonMike Koutnik - ESRIJohn Ellingson – Jackson County

WLIA Coordinate Systems Task Force

• Today’s Presentation:

- Al Vonderohe – WCCS: Redesign Objectives, Strategy, and Methodology- John Ellingson – WCCS: Testing the Redesign- Mike Koutnik – GIS applications- Ted Koch – Summary & Questions

WLIA Coordinate Systems Task Force• Mission:

• Analyze and document the foundations of the WCCS• Investigate, analyze and document software

implementations of WCCS• Investigate the redesign of the WCCS• Register WCCS with standards setting organization• Document WCCS proceedings• Develop user-focused documentation• Evaluate and make recommendations regarding

statutory changes • Present TF recommendations to WLIA Board

WLIA Coordinate Systems Task Force• Task Force Members:

• Tom Bushy ESRI• Diann Danielsen Dane County• John Ellingson Jackson County• Pat Ford Brown County• Gene Hafermann WI Dept of Transportation• David Hart UW-Madison Sea Grant• Ted Koch State Cartographer, Chair• Mike Koutnik ESRI• John Laedlein WI Dept of Natural Resources• Gerald Mahun Madison Area Technical College• David Moyer, Acting State Advisor Nat’l Geodetic Survey• Karl Sandsness Ayres Associates• Glen Schaefer WI Dept of Transportation• Jerry Sullivan WI Dept of Administration• Al Vonderohe UW-Madison, Dep’t of Civil & Environmental Engineering• Jay Yearwood City of Appleton• AJ Wortley State Cartographer’s Office

WLIA Coordinate Systems Task Force

• Task Force Accomplishments – Past Year• 6 meetings in past 12 months• Task Force decision to move ahead with redesign• WLIB directs Strategic Initiative Grant to fund redesign• Jackson County administers redesign contract• Initial redesign work is completed and tested• Various public presentations on Task Force work• Discussions on “next steps” regarding documentation &

education

From WCCS (Wisconsin County Coordinate System)

ToWISCRS (Wisconsin County Reference Systems)

Alan Vonderohe

Ellipsoid

Meridian of Longitude

Prime Meridian

Parallel of Latitude

S

N

Equator

Prim

e M

erid

ian

λP

φP

P

Latitude (φP) and Longitude (λP) of point P

Mer

idia

n of

Lon

gitu

de

Parallel of Latitude

Fundamental Descriptors of Position

Rotate about minor axis to generate oblate spheroid.

Spheroid used for current national geodetic datum (NAD83) is named “GRS 80”:• a = 6378137.0 m• b = 6356752.3141403 m

Major AxisM

inor

Axi

s a

b

a = Semi-Major Axis

b = Semi-Minor Axis

Elements of an Ellipse

Computational and Visualization Problem• Latitude / Longitude are angular, not rectangular

coordinates.• Ellipsoid surface cannot be cut and laid flat.• Latitude / Longitude must be projected to a

“developable” surface to obtain rectangular coordinates.

Developable Surfaces

Equator

Greenwich

λPφP

Equator

Greenwich

λPφP

Equator

Greenwich

λPφP

CylinderCone

Plane

One Way to Conceptualize “Projection”

Points on the ellipsoid are projected to the projection surface by straight lines from the center of the ellipsoid.

Note scale factor and how it varies across the projection surface.

Note: Some map projections are purely mathematical and have no graphical counterpart.

Ellipsoid Surface

Map Projection Surface

oa

a’

bb’

c

d’

d

e

e’

fg

g’

h’

h

Ground-to-Grid

Ellipsoid

Earth

Projection Surface

AB

A'

B'

A"

B"

Measurements are made here

Problem: Length distortion occurs when projecting from:

- Ground (Earth) to ellipsoid- Ellipsoid to projection surface

GIS spatial databases and infrastructure designs are referenced here

Ground-to-Grid• Two step process to obtain grid (map projection)

distances from ground distances:

• Or

))(( actorEllipsoidFDD groundellipsoid =

))(( rScaleFactoDD ellipsoidgrid =

))()(( rScaleFactoactorEllipsoidFDD groundgrid =

Wisconsin County Coordinates

• Original WCCS Objective:1. Make differences between ground distances and grid

distances negligible for most applications.• Original Design Strategy:

1. Restrict extent of each projection so scale factor is approximately equal to one everywhere.

2. For each projection, enlarge the ellipsoid by adding an amount that brings it to about the mean elevation of the terrain. This causes the ellipsoid factor to be approximately equal to one everywhere.

Wisconsin County Coordinates

72 Counties

59 Coordinate Systems

24 Lambert

35 Transverse Mercator

Lambert Conformal Conical Projection

Scale variation is greater north-south than east-west.

Lambert Conformal Conical Projection

Projection Parameters:

λ0 (longitude of central meridian)

φ1, φ2 (latitudes of standard parallels)

φ0, X0,Y0 (latitude, false easting, false northing of the coordinate origin)Alternative to φ1, φ2 is φ0,k0 (latitude

and scale factor at central parallel).

Standard Parallel

Standard Parallel

Central Meridian (λ0)

φ1

φ2

XoYo

X

Y

φ0X

Y

φ0

Transverse Mercator Projection

Scale variation is greater east-west than north-south.

Transverse Mercator Projection

Projection Parameters:

λ0 (longitude of central meridian)

k0 (scale factor along central meridian)

φ0, X0,Y0 (latitude, false easting, false northing of the coordinate origin)

Yo

λ0,k0

XoX

Y

φ0

Cen

tral M

erid

ian

Wisconsin County Coordinates

• Problem:• Each projection has its own ellipsoid.• This makes it seem like each projection has its own

datum.• Confusion abounds.

WLIA Task Force

• In 2004, WLIA formed the Wisconsin Coordinate Systems Task Force to address this and other spatial referencing issues.

• Ultimately, the Task Force recommended redesign of the system, established criteria, and obtained funding.

Redesign Objectives

1. Redesign the coordinate systems so there is no need to enlarge the ellipsoid.

– There will be only one ellipsoid (GRS80) for everyone.2. Redesigned coordinates should not differ by more than 5mm from

the originals anywhere on any projection.– Legacy data will be preserved.– Existing and new data can be combined without transforming either.

Redesign Strategy

1. Multiply scale factor on Central Meridian (Transverse Mercator) or Central Parallel (Lambert) by inverse of ellipsoid factor to obtain provisional scale factor.– Causes ellipsoid factor and scale factor to be approximate reciprocals of one

another, so when they are multiplied together the result is approximately equal to one.

2. Adjust false northing, false easting, and provisional scale factor to account for effects of differences of the two ellipsoids (GRS80 and enlarged).

Redesign Methodology• Methodology:

1. Use DNR statewide map to obtain boundaries for each projection.2. Generate a 0.5-mile grid of test points within a 2-mile buffer for each

projection.

Redesign Methodology• Methodology:

3. Compute provisional scale factor for each projection.4. Using provisional scale factor, compute provisional county

coordinates for each grid point.5. Compute original county coordinates for each grid point.6. Develop observation equations for each grid point:

oolprovisionaNoriginalooriginal

oolprovisionaEoriginalooriginal

NNNSvNN

EEESvEE

Δ+−=+−

Δ+−=+−

)()(

)()(

Redesign Methodology• Methodology:

7. Compute least squares solution of about 10,000 equations for each projection to obtain shifts in false northing and false easting, and multiplier for provisional scale factor.

8. Final Transverse Mercator parameters are:

Number of Transverse Mercator parameters is reduced from 7 to 5 (no need for design elevation and geoidalseparation).

ooriginaloredesignedoooriginaloredesignedo

lprovisionaoredesignedooriginaloredesignedooriginaloredesignedo

NNNEEE

Skk

Δ+=Δ+=

===

)()()()(

)()()()()()(

;

;*;; φφλλ

Redesign Methodology• Methodology:

9. Final Lambert parameters are:

• Number of Lambert parameters is reduced from 8 to 5.• φo(original) is computed from φ1(original) and φ2(original).• Coordinate origin is shifted to φo, λo.• No(original) at new coordinate origin is computed, not given.

ooriginaloredesignedoooriginaloredesignedo

lprovisionaoredesignedooriginaloredesignedooriginaloredesignedo

NNNEEE

Skk

Δ+=Δ+=

===

)()()()(

)()()()()()(

;

;*;; φφλλ

Redesign Methodology

• Methodology: 10. Compute differences between redesigned and original

coordinates at each grid point.11. Find maximum shifts in northings and eastings to

check against 5mm tolerance.12. Prepare isoline (contour) maps of coordinate shifts.

Redesign Results

• Results: • All coordinate systems meet the redesign criterion:

• All coordinate shifts are less than 5mm.• Typical coordinate shifts range from

–3mm to +3mm.• Some counties have maximum shifts of less than 1mm.• Maximum shifts are in Oneida and Vilas (Lambert) and

Ashland and Forest (Transverse Mercator).

Coordinate Shifts

Shift in Easting (mm) Shift in Northing (mm)

Buffalo County (Typical Transverse Mercator)

Coordinate Shifts

Shift in Easting (mm) Shift in Northing (mm)

Forest County (Worst-Case Transverse Mercator)

Coordinate Shifts

Shift in Easting (mm) Shift in Northing (mm)

Burnett County (Typical Lambert)

Coordinate Shifts

Shift in Easting (mm) Shift in Northing (mm)

Vilas County (Worst-Case Lambert)

Status

• Validation: • Independent testing by four individuals using various

software packages and programming techniques.• All have concluded that the redesign meets the 5mm

criterion. • Draft final report under review.

• Final submittal during March.

WISCRS (Wisconsin County Reference Systems)

• The Task Force has decided:• - To retain the name “WCCS (Wisconsin County

Coordinate System)” for the original.• - To name the redesigned “WISCRS (Wisconsin

County Reference Systems)”.• Individual county systems are suggested to be referred to

as “WISCRS, Dane County”, for example.

Wisconsin County Coordinate System

Testing the Redesign

John Ellingson, Land Information CoordinatorJackson County

CONTACT INFORMATION

• EMAIL: john.ellingson@co.jackson.wi.us• Tele: 715-284-0221

TO VIEW COORDINATE TEST DATA:• Go To: www.sco.wisc.edu• Click on: Coordinate Systems• Click on: Task Force • Click on: County Coordinate Test Point Data

(Listed under Task Force Documents)

GIS Applications

Mike KoutnikESRI

Using WISCRS inArcGIS Desktop

Using WISCRS inUsing WISCRS inArcGIS DesktopArcGIS Desktop

Prototype of access to WISCRS coordinate

systems in ArcCatalog

Building a WISCRS .prjBuilding a WISCRS .Building a WISCRS .prjprj

•• Geographic COORDYSGeographic COORDYS•• All counties use NAD 83 HARNAll counties use NAD 83 HARN

•• Projected COORDSYSProjected COORDSYS•• Unique to each countyUnique to each county

WISCRS Geographic CoordsysWISCRS Geographic WISCRS Geographic CoordsysCoordsys

•• Specified as Specified as NAD 83 HARNNAD 83 HARN

•• Same for all Same for all countiescounties

WISCRS Projected CoordsysEx: Lambert Conic

WISCRS Projected WISCRS Projected CoordsysCoordsysEx: Lambert ConicEx: Lambert Conic

•• Latitudes same for:Latitudes same for:•• Latitude of OriginLatitude of Origin•• Both standard Both standard

parallelsparallels•• Scale factors can be Scale factors can be

greater than 1greater than 1

ArcMap Projections WISCRS Dane (Foot US) to WTM 27 (meters)

ArcMap Projections ArcMap Projections WISCRS Dane (Foot US) to WTM 27 (meters)WISCRS Dane (Foot US) to WTM 27 (meters)

On-the-fly Projection

Setting thetransformation method

Questions?Questions?Questions?